Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On sampling theory associated with the resolvents of singular Sturm-Liouville problems


Author: M. H. Annaby
Journal: Proc. Amer. Math. Soc. 131 (2003), 1803-1812
MSC (2000): Primary 41A05, 34B05, 94A20
DOI: https://doi.org/10.1090/S0002-9939-02-06727-8
Published electronically: October 2, 2002
MathSciNet review: 1955268
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.


References [Enhancements On Off] (What's this?)

  • 1. M.H. Annaby and M.A. El-Sayed, Kramer-type sampling theorems associated with Fredholm integral operators, Methods Appl. Anal. 2 (1995), 76-91. MR 96h:45011
  • 2. M.H. Annaby and A.I. Zayed, On the use of Green's function in sampling theory, J. Integral Equations and Applications, 10 (1998), 117-139. MR 99g:34061
  • 3. M.H. Annaby and G. Freiling, Sampling expansions associated with Kamke problems, Math. Z. 234 (2000), 163-189. MR 2001b:34024
  • 4. M.H. Annaby and P.L. Butzer, On sampling associated with singular Sturm-Liouville eigenvalue problems: the limit-circle case, to appear.
  • 5. P.L. Butzer and G. Nasri-Roudsari, Kramer's sampling theorem in signal analysis and its role in mathematics. In: Image Processing, Mathematical Methods and Applications, The Institute of Mathematics and its Applications, New series, No 61, Clarendon Press, Oxford 1997, 49-95.
  • 6. J.A. Cochran, The Analysis of Linear Integral Equations, McGraw-Hill, New York, 1972. MR 56:6301
  • 7. E.A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. MR 16:1022b
  • 8. N. Dunford and J.T. Schwarz, Linear Operators, II, Wiley, New York, 1963. MR 32:6181
  • 9. W.N. Everitt, G. Schöttler and P. L. Butzer, Sturm-Liouville boundary value problems and Lagrange interpolation series, Rend. Math. Appl. 14 (1994), 87-126. MR 95j:34040
  • 10. W.N. Everitt, A note on the self-adjoint domains of second-order differential equations, Quart. J. Math. 13 (1963), 41-45. MR 26:1534
  • 11. C.T. Fulton, Parametrizations of Titchmarsh's ' $m(\lambda)$'-Function in The Limit-Circle Case, Doctoral thesis, RWTH-Aachen, 1973.
  • 12. C.T. Fulton, Parametrizations of Titchmarsh's $m(\lambda)$-function in the limit-circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63. MR 56:8950
  • 13. C.T. Fulton, Expansions in Legendre functions, Quart. J. Math. 33 (1982), 215-222. MR 83e:34030
  • 14. A. Haddad, K. Yao and J. Thomas, General methods for the derivation of the sampling theorems, IEEE Trans. Inf. Theory, IT-13 (1967), 227-230.
  • 15. H.P. Kramer, A generalized sampling theorem, J. Math. Phys. 38 (1959), 68-72. MR 21:2550
  • 16. N.N. Lebedev, Special Functions and Their Applications, Prentice Hall, Englewood Cliffs, N.J., 1965. MR 30:4988
  • 17. M.A. Naimark, Linear Differential Operators I, Elementary theory of linear differential operators, Frederick Ungar Publishing, New York, 1967. MR 35:6885
  • 18. I. Stakgold, Green's Functions and Boundary Value Problems, second edition, Wiley, New York, 1998. MR 99a:35002
  • 19. E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations Part I, Clarendon Press, Oxford, 1962. MR 31:426
  • 20. A.I. Zayed, G. Hinsen and P.L. Butzer, On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems, SIAM J. Appl. Math. 50 (1990), 893-909. MR 91c:94011
  • 21. A.I. Zayed, On Kramer's sampling theorem associated with general Sturm-Liouville problems and Lagrange interpolation, SIAM J. Appl. Math. 51 (1991), 575-604. MR 92b:34038
  • 22. A.I. Zayed, A new role of Green's function in interpolation and sampling theory, J. Math. Anal. Appl. 175 (1993), 222-238. MR 94d:41011
  • 23. A.I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, 1993. MR 95f:94008
  • 24. A.I. Zayed, Sampling in a Hilbert space, Proc. Amer. Math. Soc. 124 (1996), 3767-3776. MR 97b:41007

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A05, 34B05, 94A20

Retrieve articles in all journals with MSC (2000): 41A05, 34B05, 94A20


Additional Information

M. H. Annaby
Affiliation: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
Address at time of publication: Department of Mathematics, Arizona State University, P.O. Box 871804, Tempe, Arizona 85287-1804
Email: mnaby@math-sci.cairo.eun.eg, annaby@math.la.asu.edu

DOI: https://doi.org/10.1090/S0002-9939-02-06727-8
Keywords: Sampling theory, singular Sturm-Liouville problems, Green's function, resolvent kernels, Legendre and Bessel functions
Received by editor(s): November 15, 2000
Received by editor(s) in revised form: January 18, 2002
Published electronically: October 2, 2002
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society